\begin{frame}[allowframebreaks]
\frametitle{Wang definitions}

\begin{define}[Labeled Wang tiles \cite{LonatiPradella2011}]

A labeled Wang tile is an element $A = (a, t, l, r, b) \in \Sigma\times C^4$,
where 
\begin{itemize}
	\item t, b, r, l represent the colors at top, bottom, right and left edges
	\item $\Sigma$ is a finite alphabet
	\item C is a set of colors, with the special color $\#$ representing
	borders
\end{itemize}

\end{define}

For better readability, we represent labeled Wang tiles as 
\setlength{\tabcolsep}{2pt}
\(A = \begin{tabular}{rcl}
		  & t         &   \\[-0.5ex]
		l & \boxed{a} & r \\[-0.2ex]
		  & b         & 
	  \end{tabular}\).
\setlength{\tabcolsep}{6pt}

\begin{itemize}
	\item $A_d$ is the color of the edge of A towards direction $d \in \{\leftarrow, \rightarrow, \uparrow, \downarrow\}$
	\item $\Sigma_{4C}$ represent the set of tiles with labels in $\Sigma$ and colors in
$C$
	\item $\Delta_A$ is the domain of $A$ and represent the set of
defined colors.
\end{itemize} 

\begin{define}[Wang pictures \cite{LonatiPradella2011}]
A picture $p\in\Sigma_{4C}^{**}$ is called a Wang picture if
\begin{itemize}
  \item all borders are colored with $\#$
  \item all non-borders are not colored with $\#$
  \item $p(i,j)_\rightarrow = p(i, j + 1)_\leftarrow$ for every $1 \leq j$
  \textless{} $l_1(p)$
  \item $p(i,j)_\downarrow = p(i + 1, j)_\uparrow$ for every $1 \leq i$
  \textless{} $l_2(p)$
\end{itemize}

The label of a Wang picture $P$ over $\Sigma_{4K}$ is the picture having for
pixels the labels of pixels of P.
Any (partial) Wang picture is called a (partial) Wang tiling of its label.

\end{define}

\pagebreak

If w is a (2,2) Wang picture, we will use one of the following representation: 

\begin{center}
\begin{tabular}{|c|c|c|}
\hline
wang picture & label & short form \\
\hline
\(
\setlength{\tabcolsep}{2pt}
\begin{tabular}{|c|c|}
\hline
		\begin{tabular}{rcl}
		   & \#        &   \\[-0.5ex]
		\# & \boxed{a} & 4 \\[-0.2ex]
		   & 1         & 
		\end{tabular}  			 	&		\begin{tabular}{rcl}
											  & \#         &    \\[-0.5ex]
											4 & \boxed{b}  & \# \\[-0.2ex]
											  & 3          & 
											\end{tabular} 				\\		
\hline
		\begin{tabular}{rcl}
		   & 1         &   \\[-0.5ex]
		\# & \boxed{b} & 2 \\[-0.2ex]
		   & \#        & 
		\end{tabular} 				&	 	\begin{tabular}{rcl}
											  & 3         &    \\[-0.5ex]
											2 & \boxed{a} & \# \\[-0.2ex]
											  & \#        & 
											\end{tabular}				\\
\hline
\end{tabular}
\setlength{\tabcolsep}{6pt}
\)
& 
\(
\setlength{\tabcolsep}{2pt}
\begin{tabular}{|c|c|}
\hline
		a  			 	&		b 				\\		
\hline
		b 				&	 	a				\\
\hline
\end{tabular} \) 
\setlength{\tabcolsep}{6pt}
&
\( 
\setlength{\tabcolsep}{2pt}
\begin{tabular}{|rcccl|}
\hline
   & \#        &   & \#        &    \\[-0.5ex]
\# & \boxed{a} & 4 & \boxed{b} & \# \\[-0.2ex]
   & 1         &   & 3         &    \\[-0.5ex]
\# & \boxed{b} & 2 & \boxed{a} & \# \\[-0.2ex]
   & \#        &   & \#        &    \\
\hline
\end{tabular}
\setlength{\tabcolsep}{6pt}
\)  \\
\hline
\end{tabular}
\end{center}

\begin{define}[Wang system \cite{LonatiPradella2011}]
A Wang system is a tripel $\omega = (C, \Sigma, \Theta)$, where
\begin{itemize}
  \item C is a set of colors
  \item $\Sigma$ is a finite alphabet
  \item $\Theta \subseteq \Sigma \times C^4$ a set of tiles. 
\end{itemize}
\end{define}

The language generated by $\omega$ is the language $L(\omega) \subseteq
\Sigma^{**}$ of the labels of all Wang pictures built with tiles in $\Theta$.

\end{frame}

\begin{frame}[allowframebreaks]
\frametitle{Example}

\setlength{\tabcolsep}{2pt}
\begin{Example}
\begin{align*} \Theta = \left\lbrace
\begin{tabular}{c}
\begin{tabular}{rcl}
 & \# & \\[-0.5ex]
\# & \boxed{x} & - \\[-0.2ex]
 & x & 
\end{tabular}, 
\begin{tabular}{rcl}
 & \# & \\[-0.5ex]
- & \boxed{x} & - \\[-0.2ex]
 & x & 
\end{tabular}, 
\begin{tabular}{rcl}
 & \# & \\[-0.5ex]
- & \boxed{x} & \# \\[-0.2ex]
 & x & 
\end{tabular}, 
\begin{tabular}{rcl}
 & \# & \\[-0.5ex]
\# & \boxed{x} & \# \\[-0.2ex]
 & x & 
\end{tabular}\\[1ex]
\begin{tabular}{rcl}
 & x & \\[-0.5ex]
\# & \boxed{y} & - \\[-0.2ex]
 & x & 
\end{tabular}, 
\begin{tabular}{rcl}
 & x & \\[-0.5ex]
- & \boxed{y} & - \\[-0.2ex]
 & x & 
\end{tabular}, 
\begin{tabular}{rcl}
 & x & \\[-0.5ex]
- & \boxed{y} & \# \\[-0.2ex]
 & x & 
\end{tabular}, 
\begin{tabular}{rcl}
 & x & \\[-0.5ex]
\# & \boxed{y} & \# \\[-0.2ex]
 & x & 
\end{tabular} \\[1ex]
\begin{tabular}{rcl}
 & x & \\[-0.5ex]
\# & \boxed{x} & - \\[-0.2ex]
 & \# & 
\end{tabular}, 
\begin{tabular}{rcl}
 & x & \\[-0.5ex]
- & \boxed{x} & - \\[-0.2ex]
 & \# & 
\end{tabular}, 
\begin{tabular}{rcl}
 & x & \\[-0.5ex]
- & \boxed{x} & \# \\[-0.2ex]
 & \# & 
\end{tabular}, 
\begin{tabular}{rcl}
 & x & \\[-0.5ex]
\# & \boxed{x} & \# \\[-0.2ex]
 & \# & 
\end{tabular}
\end{tabular}
\right\rbrace \\
\text{for } x,y \in \{a, b\}
\end{align*}
\end{Example}
\setlength{\tabcolsep}{6pt}

$\omega = (\{a, b, -\}, \{a, b\}, \Theta)$. Then $L(\omega) = \{p \in \{a, b\}^{**} \mid p(1, i) = p(l_1(p), i) \text{ for } 1 \leq i \leq l_2(p)\}$ is the language, which contains only pictures, where the first and last row are equal. 

\begin{center}
\begin{tabular}{rcl}
\begin{tabular}{rcccccccl}
   & \# & & \# & & \# & & \# & \\
\# & \boxed{a} & - & \boxed{a} & - & \boxed{b} & - & \boxed{b} & \#\\
 & $a$ & - & $a$ & - & $b$ & - & $b$ & \\
 \# & \boxed{b} & - & \boxed{a} & - & \boxed{b} & - & \boxed{a} & \#\\
 & $a$ & - & $a$ & - & $b$ & - & $b$ & \\
 \# & \boxed{a} & - & \boxed{b} & - & \boxed{b} & - & \boxed{b} & \#\\
 & $a$ & - & $a$ & - & $b$ & - & $b$ & \\
 \# & \boxed{a} & - & \boxed{a} & - & \boxed{b} & - & \boxed{b} & \#\\
   & \# & & \# & & \# & & \# & 
\end{tabular}
&
$\leadsto$
&
\boxed{\begin{tabular}{rccl}
a & a & b & b\\
b & a & b & a\\
a & b & b & b\\
a & a & b & b
\end{tabular}}
\end{tabular}
\end{center}

\end{frame}